ОЦЕНКА ГЕССИАНОВ ОГРАНИЧЕННЫХ m-ВЫПУКЛЫХ ФУНКЦИЙ
Respectfully dedicated to the 80th anniversary of Shavkat Arifzhanovich Alimov and the 70th
anniversary of Ravshan Radzhabovich Ashurov in recognition of their outstanding contributions
to science and education.
A novel approach to studying m-convex (m − cv) functions has been developed by mathematicians
from the Khorezm branch of the V.I. Romanovskiy Institute of Mathematics of the Academy of
Sciences of Uzbekistan and the National University of Uzbekistan named after Mirzo Ulugbek. This
research method is based on the connection between m-convex functions and m-subharmonic (sh m )
functions. Through this established connection, it was shown that H k (u), k = 1,2,...,n − m +
1, can be defined as Borel measures within the class of bounded m − cv functions. Several basic
properties of these measures were also proven.
In the present work, more significant properties of such measures are established, including uniform
estimates for the integral mean values of the Hessians H k (u), k = 1,2,...,n − m + 1, within the
class of bounded m − cv functions.
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